RRJoST(2013) 1-10 © STM Journals 2013. All Rights Reserved Page 1 Research & Reviews: Journal of Statistics ISSN: 2278–2273 www.stmjournals.com Construction of Neighbour Balanced BIBRC Designs Aatish Kumar Sahu 1 *, Anurup Majumder 2 1 Department of Agricultural Economics, SASRD, Nagaland University, Medziphema, Nagaland-797106 2 Department of Agricultural Statistics, Bidhan Chandra Krishi Viswavidyalaya, Mohanpur, Nadia, W. Bengal-741252 Abstract Construction of a series of Balance Incomplete Block with nested Rows and Columns (BIBRC) designs having Neighbor Balanced properties has been presented in this paper. The series of BIBRC designs has been developed on the concept of mutually orthogonal mates (MOM’s) of existing BIBRC designs with v as a prime number. The designs thus constructed are found to be first order neighbor balanced with respect to rows, columns as well as diagonally, with every pair of treatment occurred in immediate neighboring plots n r times in rows, n c times in columns and n d times in diagonals respectively. Keywords: BIBRC designs, Neighbor Balanced BIBRC designs, OM and MOM of BIBRC designs, Galois Field (v), Mod (v), Symmetric neighbor differences. Author for Correspondence E-mail: aatishksahu@gmail.com INTRODUCTION In agricultural experiments, the neighbor plot effects of treatments can be guarded by using optimal neighbor balanced block designs. Neighbour balanced designs were initially used in serology by Rees [1] for use in virus research. Kiefer and Wynn [2] proved that a neighbour balanced Balanced Incomplete Block with nested Rows-Columns (BIBRC) design would be highly efficient within the class of binary block designs under the fixed effects (row, column and block). Two- dimensional designs balanced for nearest neighbours in rows, in columns and in diagonals with no like neighbours in any of these directions were first constructed by Gill and Shukla [3] using a set of v Latin Squares. However, systematic methods for constructing such designs with single and multiple blocks were first developed by Morgan and Uddin [4]. A useful discussion on the history of two- dimensional neighbour balanced designs that allows like neighbours in diagonal direction can be found in Morgan [5]. Martin and Eccleston [6] described a construction technique based on semi- balanced array that results in BIBRCs with excellent neighbor balanced properties having strong optimality properties within the class of block designs. Srivastava and Morgan [7] developed (2×2) blocks of BIBRCs, in which each of the row, column and block component design is a balanced incomplete block design, satisfying neighbour balanced properties. Various types of two-dimensional nearest neighbour balanced designs are introduced and studied in Afsarinejad and Seeger [8], Freeman [9,10], Morgan [5], Morgan and Uddin [11], Street [12] and Street and Street [13]. Uddin [14] defined the neighbour balanced BIBRC design with one additional condition of diagonally balanced property to Singh and Dey [15], and developed two methods by which two series of neighbour balanced BIBRC designs can be constructed by defining neighbor balanced in rows and columns combined, and balanced for neighbours in diagonals. Uddin [16] showed some series of previously known balanced incomplete block designs with nested rows and columns to be balanced for neighbours in rows, columns and in diagonals, when blocks are assumed to be on toruses and/or cylinders. These series of designs with planar blocks are found to be highly efficient within the class of binary